To convert r = 3 cos 2θ into rectangular coordinates, we use the fact that

cos 2θ = cos²θ - sin²θ.

So r = 3 cos 2θ = 3(cos²θ - sin²θ).

Now since , and r² = x² + y², we have

and

Taking the positive square root of r² = x² + y² gives us:

So r = 3 cos 2θ in polar coordinates is equivalent to

in rectangular coordinates.

We see that our equation in polar coordinates, r = 3 cos 2θ, is much simpler than the rectangular equivalent.